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- Cantor distribution

The **Cantor distribution** is the probability distribution whose cumulative distribution function is the Cantor function.

This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.

Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning.

The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:

*\begin{align}
**C*_{0}=*{}**&**[*0*,*1*]**\\[*8*pt]
**C*_{1}=*{}**&**[*0*,*1*/*3*]\cup[*2*/*3*,*1*]**\\[*8*pt]
**C*_{2}=*{}**&**[*0*,*1*/*9*]\cup[*2*/*9*,*1*/*3*]\cup[*2*/*3*,*7*/*9*]\cup[*8*/*9*,*1*]**\\[*8*pt]
**C*_{3}=*{}**&**[*0*,*1*/*27*]\cup[*2*/*27*,*1*/*9*]\cup[*2*/*9*,*7*/*27*]\cup[*8*/*27*,*1*/*3*]\cup**\\[*4*pt]
**{}**&**[*2*/*3*,*19*/*27*]\cup[*20*/*27*,*7*/*9*]\cup[*8*/*9*,*25*/*27*]\cup[*26*/*27*,*1*]**\\[*8*pt]
**C*_{4}=*{}**&**[*0*,*1*/*81*]\cup[*2*/*81*,*1*/*27*]\cup[*2*/*27*,*7*/*81*]\cup[*8*/*81*,*1*/*9*]\cup[*2*/*9*,*19*/*81*]\cup[*20*/*81*,*7*/*27*]\cup**\\[*4*pt]
**&**[*8*/*27*,*25*/*81*]\cup[*26*/*81*,*1*/*3*]\cup[*2*/*3*,*55*/*81*]\cup[*56*/*81*,*19*/*27*]\cup[*20*/*27*,*61*/*81*]\cup**\\[*4*pt]
**&**[*62*/*81*,*21*/*27*]\cup[*8*/*9*,*73*/*81*]\cup[*74*/*81*,*25*/*27*]\cup[*26*/*27*,*79*/*81*]\cup[*80*/*81*,*1*]**\\[*8*pt]
**C*_{5}=*{}**&* … *
\end{align}
*

The Cantor distribution is the unique probability distribution for which for any *C*_{t} (*t* ∈ ), the probability of a particular interval in *C*_{t} containing the Cantor-distributed random variable is identically 2^{−t} on each one of the 2^{t} intervals.

It is easy to see by symmetry that for a random variable *X* having this distribution, its expected value E(*X*) = 1/2, and that all odd central moments of *X* are 0.

The law of total variance can be used to find the variance var(*X*), as follows. For the above set *C*_{1}, let *Y* = 0 if *X* ∈ [0,1/3], and 1 if *X* ∈ [2/3,1]. Then:

*\begin{align}
\operatorname{var}(X)**&*=*\operatorname{E}(\operatorname{var}(X\mid**Y))*+*\operatorname{var}(\operatorname{E}(X\mid**Y))**\\
**&*=

1 | |

9 |

*\operatorname{var}(X)*+*\operatorname{var}
**\left\{
**\begin{matrix}*1*/*6*&*withprobability 1*/*2*\* *
*5*/*6*&*withprobability 1*/*2*
**\end{matrix}
**\right\}**\\
**&*=

1 | |

9 |

*\operatorname{var}(X)*+

1 | |

9 |

*\end{align}
*

From this we get:

\operatorname{var}(X)= | 1 |

8 |

*.*

A closed-form expression for any even central moment can be found by first obtaining the even cumulants^{[1]}

*\kappa*_{2n}=

2^{2n-1}(2^{2n}-1)B_{2n} | |

n(3^{2n}-1) |

*,**
*

where *B*_{2n} is the 2*n*th Bernoulli number, and then expressing the moments as functions of the cumulants.

- Book: E. . Hewitt . K. . Stromberg . Real and Abstract Analysis . registration . Springer-Verlag . Berlin-Heidelberg-New York . 1965.
*This, as with other standard texts, has the Cantor function and its one sided derivates.* - News: Tian-You . Hu . Ka Sing . Lau . Fourier Asymptotics of Cantor Type Measures at Infinity . Proc. A.M.S. . 130 . 9 . 2002 . 2711–2717.
*This is more modern than the other texts in this reference list.* - Book: Knill, O. . Probability Theory & Stochastic Processes . Overseas Press . India . 2006.
- Book: Mattilla, P. . Geometry of Sets in Euclidean Spaces . Cambridge University Press . San Francisco . 1995.
*This has more advanced material on fractals.*

- Web site: Morrison . Kent . Random Walks with Decreasing Steps . Department of Mathematics, California Polytechnic State University . 1998-07-23 . 2007-02-16 . 2015-12-02 . https://web.archive.org/web/20151202055102/http://www.calpoly.edu/~kmorriso/Research/RandomWalks.pdf . dead .